River Crossing: The intent was to cross perpendicular to the current. The phrasing was bad.

Given a position formula:

Which has units of time?

Tuesday, February 25, 2020 3:17 PM

11. Mass and Force Page 1

Slowing to a stop

11. Mass and Force Page 2

Dynamics is about the causes and control of motion.

Newton's 1st Law: Inertia. An object at rest tends to stay at rest, and an object in motion tends to stay in motion at constant velocity, except when affected by a net force.

Any push, pull, or drag.•

Forward force tries to speed up motion.○

Backward force tries to slow down motion.○

Sideways force tries to change direction of motion.○

Any attempt to affect the motion.•

The success of a force's attempted effect depends on other forces.•

Force is the basic mechanical interaction between objects.

Newton's 2nd Law: Mathematical description of effect of forces.

Net force or total force is what actually matters.•Forces are vectors.•Acceleration is the actual result of net force.•Mass (m) measured in kilograms (kg), is the "difficulty" of causing accel.•

Force is measured in newtons (N), which is an abbreviation.

Newton's Third Law: Every force is really an interaction between two objects, and the two objects are acted upon by opposite-direction copies of the force.

The force on the left object is leftward.

DynamicsTuesday, February 25, 2020 3:59 PM

11. Mass and Force Page 3

The object•All forces affecting the object•Construction lines•Coordinate system•Labels•

FBD shows:

F_N is the "normal force" which is the result of contact between solid objects. It's perpendicular to the surface of contact.

The scale actually measures the normal force.The gravitational force is proportional to the mass.

So now we get a very common result:

There is a normal force.•Zero acceleration•No other force with y-component.•Normal force is actually upward.•

What needs to be true to keep this result?

Even though F_N = m g is very common, but keep your eye out for the exceptions that can break this relationship.

Free Body DiagramsTuesday, February 25, 2020 4:23 PM

11. Mass and Force Page 4

How do we make sure to include all forces affecting our object?

Two categories: Distance and contact

Gravity•Electromagnetic•

Distance forces:

Normal force - pushing away from surface○

Friction force - dragging along surface○

Solid contact:•

Tension from string, rope, chain, or wire•

Static pressure force○

Drag or resistance force○

Buoyancy○

Fluid forces•

Contact forces:

Ex: Object hanging from a structural frame

At this point, we have implemented Newton's 3rd Law.

Effectively, the normal force must hold up the weight of both objects, combined.

Force MechanismsThursday, February 27, 2020 3:16 PM

12. More Forces Page 5

What happens if this system is in an accelerating elevator?

Upward accel is easy with some extra normal force.What about downward acceleration? Decreased normal force.How much downward acceleration is allowed?

The acceleration is allowed to be negative, but only down to -9.8 m/s^2.

12. More Forces Page 6

What is the maximum angle I could use for this force and maintain contact?

Above that angle, there would be y-acceleration as well.

Dynamic situationsThursday, February 27, 2020 4:05 PM

12. More Forces Page 7

Typical: Solve for F_N and a.

When the object gets to the bottom, there is a vertical wall. After it comes to rest, what is the force from the wall?

Inclined PlaneThursday, February 27, 2020 4:17 PM

12. More Forces Page 8

Friction is the tangential part of the contact force.Friction opposes any sliding or attempted sliding.

Let's do a thought experiment. Imagine ramping up the horizontal pulling force on a chair.

How much horizontal force is needed to get the crate moving?1.Ex: A 20 kg crate is sitting on a horizontal floor. The friction coefficients are and .

What happens if I only exert ?2.

What happens if I exert for 3.0 s?3.

FrictionTuesday, March 3, 2020 3:11 PM

13. Friction Page 9

What happens if I now stop pulling?4.

What is this acceleration measured as a fraction of g?

The coefficient of friction is the max acceleration that a car/person can get on their own, measured in terms of g.

13. Friction Page 10

What direction of pulling force will minimize the amount of force necessary to start the object moving from rest.

I could take the derivative and set it to zero, in order to find the theta that minimizes F_P, but it's easier to maximize the denominator.

I want the angle that is the inverse tangent of the static friction coefficient.

The conceptual result is that an upward pulling force makes it easier to drag the object.

Friction with a diagonal pullTuesday, March 3, 2020 4:02 PM

13. Friction Page 11

What angle will allow the object to slide downhill at constant velocity?

We can "measure" the static friction coefficient by performing an experiment: Increase the incline until the object starts to move.

Friction on an inclineTuesday, March 3, 2020 4:18 PM

13. Friction Page 12

An ideal pulley has no friction and no mass.•A pulley is a device that changes the direction of tension and connects the motion of different objects.

Half-Atwood Machine

Typically, we eliminate F_T as a variable by adding the "forward" equations:

The two remaining forces are the external forces, along the direction of motion.

Ex: If m1=0.1 kg, and m2=0.2 kg, with no friction, what is the acceleration?

To find tension, look at just m2:

PulleysThursday, March 5, 2020 3:20 PM

14. Pulleys Page 13

If there is enough static friction to keep the system at rest, what is the tension?

Threshold question: How much friction is needed?

With a more usual friction coefficient of , what are the accel and tension?

14. Pulleys Page 14

Since is always less than , gravity is always being multiplied by a small fraction, so the acceleration of an Atwood machine is slower than 9.8 m/s^2.

Note that m2 is accelerating downward while m1 is accelerating upward.This means the tension is in between the two weights.

Atwood's MachineThursday, March 5, 2020 4:14 PM

14. Pulleys Page 15

The string gets to apply its tension to the person twice.

If the person weighs 1000 N, the tension only has to be 500 N.

Often, the rope can pass through multiple pulleys at each end. Ex:

Atwood Machine with Mechanical Advantage

Pulley systems with mechanical advantage.Thursday, March 5, 2020 4:22 PM

14. Pulleys Page 16

Each week, I'll add an item to the Remote Lecture Materials

Since Exam 1, leading up to Exam 2:

Administrative QuestionsTuesday, March 24, 2020 3:32 PM

15. Centripetal Accel Page 17

HW4 will be posted soon, due Sun April 5.

Exam2 will be on Blackboard. Open book with a 75 minute time limit.Probably I'll give a 3-day window to take the exam.

15. Centripetal Accel Page 18

Perpendicular to the velocity and to the path.•Points "inward" toward the inside of a curve.•Has a particular value:•

The centripetal acceleration is:

Flat bend•Banked curve•

Two types of bends in a road:

Flat curve example: Entrance to campus

What forces contribute toward the net force that causes this acceleration?

How much grip (friction coefficient) is needed to navigate this bend?

Centripetal AccelerationTuesday, March 24, 2020 3:44 PM

15. Centripetal Accel Page 19

How much grip (friction coefficient) is needed to navigate this bend?

Dry road: ~0.7•Wet road: ~0.1 to 0.3•

That tells us the minimum "grip" of the road.

The other version of the inequality tells us the max speed.

The centripetal acceleration is horizontal (perpendicular to gravity).•Banked curve:

More, because dividing by value < 1.0•

Part balances gravity.○

Part provides acceleration.○

More, because F_N is performing double-duty.•

Is this more or less than mg?

(Answer to online chat question): Yes, Temperature can affect the friction coefficient.This depends on the materials involved.

(Answer to inline chat question): No, I won't necessarily say to use centripetal acceleration. It is applicable whenever an object is changing direction or following a curved path.

(0.25) * (81) * (9.8) = 198.45

v <= sqrt(198.45) = 14.1 m/s

15. Centripetal Accel Page 20

This tells us the velocity necessary to navigate the bend without friction.What happens if we go too fast? (we'll fly off the outside, top end of the bank)What happens if we go too slow? (we'll slide off the inside of the bend)

Static friction involves some "settling into the grooves" of a surface.•Kinetic friction involves "skipping off the ridges" of a surface.•Static friction coefficient is always at least as strong as kinetic friction coefficient.•

Q: What is the difference between and ?

15. Centripetal Accel Page 21

Structured lesson on our playlists: MW Recordings and TR Recordings

From <http://faculty.tamucc.edu/jspirko/Phys2425/>

HW4a (centripetal) and HW4b (Buoyancy) are posted.

Buoyancy is an upward force caused by being immersed in a fluid.It's equal to the weight of the displaced fluid.

(Aside: How will tests work? On Blackboard, open-book, time-limited, but flexible window to take it.)With the limited time, you won't be able to Google answers and finish the exam.

We're always using force diagrams to relate buoyancy with gravity.

Three typical force diagrams with buoyancy:

Ex: A block of styrofoam has a density of only 10% of water's density.How much of the styrofoam sticks above the surface when it floats?

With a sunken object, we're usually concerned about the third force, tension or normal force.

Buoyancy and Archimedes PrincipleThursday, March 26, 2020 3:10 PM

16. Buoyancy Page 22

With a sunken object, we're usually concerned about the third force, tension or normal force.

Ex: Gold has a density of 19 times that of water. If a 40 N hunk of gold is immersed in water, what is the tension required to hold it up?

Effect of cargo on a boat:

The weight of the cargo causes a change in buoyant force, which requires a change in displaced volume, which is observable by a change in depth.

A typical river barge is 200 ft by 35 ft.

If 10 cars, 3000 kg each, are loaded.How much does the barge subside in the water?

16. Buoyancy Page 23

Torque (Video: https://youtu.be/cpttjk7BxXA, PDF: Link)•Static Equilibrium (Video: https://youtu.be/Fw3yg7XO1Lo, PDF: Link)•

Today, you should learn about torque.

They are also at the top of your YouTube playlist: MW Recordings and TR Recordings.All 3 parts of HW4 are now posted.

Lab 6 is due FridayLab 7 is posted. Quiz due Friday 8pm. Lab due next week.

Exam 2 info posted on Remote Lecture Materials area.

There is an eXtra Credit HW posted in the Remote Lecture Materials area.The task is simple, combine a hand drawing and some equations into a PDF.

AdministrativeTuesday, March 31, 2020 3:20 PM

17. Torque Page 24

The cosine of the angle is used to calculate the component in the "adjacent direction". If the angle is measured from the x-axis, then the x-component involves the cosine.

Taking components of VectorsTuesday, March 31, 2020 3:43 PM

17. Torque Page 25

Continue to stay away from other people, especially those who have traveled from outside your area.

Cool simulation video: Simulating an epidemic

Exponential growth and epidemicsTuesday, March 31, 2020 4:01 PM

17. Torque Page 26

Easiest way to calculate torque of a force:

Leaning Ladder Problem

Pinned point? Use that.•Free object? Use Center of Mass.•Static object? Pick anywhere you want.•

To analyze torques, must pick a pivot point.

Since we had only one unknown force ( ), this result allows us to solve for that force.Other force analysis:

Static grip condition:

This is the useful result of analysis that we can use to find answers.

Torque ExamplesThursday, April 02, 2020 3:22 PM

18. Torque Examples Page 27

Exam 2 will be on Blackboard, to be taken between Wed 8am and Thu midnight.Questions will be multiple choice and numerical fill-in-the-blank or answer.No lockdown browser or proctor software. Open book. Use your equation sheet!Time limit of 60 minutes. No time to re-learn or browse the book.No communication with other people or Q&A websites during the exam.

HW4c, #5

Can solve for x from here. Easy way: Assume .

Sometimes when the algebra is awkward, there can be an advantage to choosing a different pivot point.

ReviewTuesday, April 07, 2020 3:24 PM

19. Review Page 28

This still combines with the same Newton's Second Law component equations from above, and the grip condition.This algebra is probably a little easier.

Buoyancy HW, #3

19. Review Page 29